• r )08 Sir G. Greenhill on the 



7. Clifford's definition 



c„^) = (-i)» d ^=2(-,o s /n(« + /-)ni . (i) 



is generalised for fractional values of n by interpreting lira 

 to mean Gauss's Gamma Function T(ra-l-l). 



When n is half an odd integer, the Clifford functions 

 are the differentiations or integrations of cos (2 */x + «?), 

 changing to a corresponding hyperbolic function for 

 negative values of x. 



Then we can put 



C!i/V> - sin(2i/.i? + 6) 



(UO)= -cos(2v^ + e) . . . .' (2) 



., , N cos(2vAi< + e) sin(2v/.r + e) 

 C|W =-- Tm + ~~ i5 ' 



■C_ a (#) = — V .i' cos (2 t/a? 4- e) i- J sin (2 v^.r + e), = xzCz(x). ( 3) 



_ , . ' d n , . 3 cos (2 \/x + e) 3 — 4a? . 



•C_,(«) =j'G. | (.,)<^, = «*C,(«), (4) 



and so on. 



Here the arbitrary e is at disposal, to give the two 

 separate solutions, corresponding to C and D when ra is 

 integral ; and C 1+ a( — x) is obtained by a change of the 

 circular into the hyperbolic function. 



Vibration of an elastic sphere. 



8. As an application of these functions of order h;ilf 

 an odd integer, take the D.E. (Love, ' Elasticity ') 



dru du , 2 v 



dr 2 * dr~^ 1 ?Vm = °' .... (1) 



no 

 for U = ru sin ra£, the radial displacement, with p 2 = 



k + pi 



